Abstract: Frown is an LALR(k) parser generator for Haskell 98 written in Haskell 98.

Its salient features are:

• The generated parsers are time and space efficient. On the downside, the parsers are quite large.
• Frown generates four different types of parsers. As a common characteristic, the parsers are genuinely functional (ie ‘table-free’); the states of the underlying LR automaton are encoded as mutually recursive functions. Three output formats use a typed stack representation, one format due to Ross Paterson (code=stackless) works even without a stack.
• Encoding states as functions means that each state can be treated individually as opposed to a table driven-approach, which necessitates a uniform treatment of states. For instance, look-ahead is only used when necessary to resolve conflicts.
• Frown comes with debugging and tracing facilities; the standard output format due to Doaitse Swierstra (code=standard) may be useful for teaching LR parsing.
• Common grammatical patterns such as repetition of symbols can be captured using rule schemata. There are several predefined rule schemata.
• Terminal symbols are arbitrary variable-free Haskell patterns or guards. Both terminal and nonterminal symbols may have an arbitrary number of synthesized attributes.
• Frown comes with extensive documentation; several example grammars are included.

Furthermore, Frown supports the use of monadic lexers, monadic semantic actions, precedences and associativity, the generation of backtracking parsers, multiple start symbols, error reporting and a weak form of error correction.

Contents

• Chapter 1  Introduction
  • 1.1  Obtaining and installing Frown
  • 1.2  Reporting bugs
  • 1.3  License
  • 1.4  Credits
• Chapter 2  Quick start
• Chapter 3  Tour de Frown
  • 3.1  Preliminaries: monads
  • 3.2  Basic features
    • 3.2.1  Pure grammars
    • 3.2.2  Attributes
    • 3.2.3  Interfacing with a lexer
    • 3.2.4  Monadic actions
    • 3.2.5  Backtracking parsers
    • 3.2.6  Precedences and associativity
    • 3.2.7  Multiple start symbols
    • 3.2.8  Monadic attributes
  • 3.3  Error reporting and correction
    • 3.3.1  Monadic lexers
    • 3.3.2  Error reporting
    • 3.3.3  Expected tokens
    • 3.3.4  Error correction
  • 3.4  Advanced features
    • 3.4.1  Rule schemes
    • 3.4.2  A second look at terminal symbols
    • 3.4.3  Look-ahead
    • 3.4.4  Debugging and tracing
    • 3.4.5  Output formats and optimizations
• Chapter 4  Tips and tricks
  • 4.1  Irrefutable patterns
  • 4.2  Inherited attributes
  • 4.3  Dealing with conflicts
  • 4.4  Multiple attributes
• Chapter 5  Reference manual
  • 5.1  Lexical syntax of Frown
  • 5.2  Syntax of Frown
  • 5.3  Predefined schemes
    • 5.3.1  Optional elements
    • 5.3.2  Repetition of elements
    • 5.3.3  Repetition of elements separated by a second element
    • 5.3.4  Repetition of possibly empty elements separated by a second element
  • 5.4  Output formats
  • 5.5  Invocation and options

Chapter 1  Introduction

Frown is an LALR(k) parser generator for Haskell 98 written in Haskell 98.

The work on Frown started as an experiment in generating genuinely functional LR parsers. The first version was written within three days—yes, Haskell is a wonderful language for rapid prototyping. Since then Frown has gone through several cycles of reorganization and rewriting. It also grew considerably: dozens of features were added, examples were conceived and tested, and this manual was written. In the end, Frown has become a useable tool. I hope you will find it useful, too.

1.1  Obtaining and installing Frown

Obtaining Frown

The parser generator is available from

http://www.informatik.uni-bonn.de/~ralf/frown.

The bundle includes the sources and the complete documentation (dvi, ps, PDF, and HTML).

Requirements

You should be able to build Frown with every Haskell 98-compliant compiler. You have to use a not too ancient compiler as there have been some changes to the Haskell language in Sep. 2001 (GHC 5.02 and later versions will do).

The Haskell interpreter Hugs 98 is needed for running the testsuite.

Various tools are required to generate the documentation from scratch: lhs2TeX, L^AT_EX, functional , H^EV^EA and H^AC^HA. Note, however, that the bundle already includes the complete documentation.

Installation

Unzip and untar the bundle. This creates a directory called Frown. Enter this directory.

The documentation resides in the directory Manual; example grammars can be found in Examples and Manual/Examples (the files ending in .g and .lg).

You can install Frown using either traditional makefiles or Cabal.

Using makefiles

Optionally, edit the Makefile to specify destinations for the binary and the documentation (this information is only used by make install). Now, you can trigger

which compiles Frown generating an executable called frown (to use Frown you only need this executable). Optionally, continue with

to install the executable and the documentation.

For reference, here is a list of possible targets:

make

Compiles Frown generating an executable called frown (to use Frown you only need this executable).

make install

Compiles Frown and installs the executable and the documentation.

make test

Runs the testsuite.¹

make man

Generates the documentation in various formats (dvi, ps, PDF, and HTML).

make clean

Removes some temporary files.

make distclean

Removes all files except the ones that are included in the distribution.

Using Cabal

Alternatively, you can build Frown using Cabal (version 1.1.3 or later), Haskell’s Common Architecture for Building Applications and Libraries.

For a global install, type:

If you want to install Frown locally, use (you may wish to replace $HOME by a directory of your choice):

Usage

The call

displays the various options. For more information consult this manual.

1.2  Reporting bugs

Bug reports should be send to Ralf Hinze (ralf@cs.uni-bonn.de). The report should include all information necessary to reproduce the bug: the compiler used to compile Frown, the grammar source file (and possibly auxiliary Haskell source files), and the command-line invocation of Frown.

Suggestions for improvements or request for features should also be sent to the above address.

1.3  License

Frown is distributed under the GNU general public licence (version 2).

1.4  Credits

Frown wouldn’t have seen the light of day without the work of Ross Paterson and Doaitse Swierstra. Ross invoked my interest in LR parsing and he devised the --code=stackless and --code=gvstack output formats. Doaitse invented the --code=standard format, which was historically the first format Frown supported.

A big thank you goes to Andres Löh and Ross Paterson for various bug reports and suggestions for improvement.

1

There are some known problems. The format code=stackless behaves differently for Loop.g (the generated parser is less strict than the standard one). Also, Empty.g does not work yet. Finally, error reports may differ for different formats and for optimized and unoptimized versions (as some parsers perform additional reductions before an error is reported).

Chapter 2  Quick start

First install Frown as described in Sec. 1.1. Then enter the directory QuickStart.

The file Tiger.lg, listed in Fig. 2.1, contains a medium-sized grammar for an imperative language.

---

A grammar file consists of two parts: the specification of the grammar, enclosed in special curly braces, and Haskell source code. The source file typically starts with a Haskell module header.

>  module Tiger where >  import Lexer >  import Syntax >  import Prelude hiding (exp) >  >  %{

The grammar part begins here. A context-free grammar consists of sets of terminal and nonterminal symbols, a set of start symbols, and set of productions or grammar rules. The declaration below introduces the terminal symbols. Each terminal is given by a Haskell pattern of type Terminal.

>  Terminal  =  DO  |  ELSE  |  END   |  FUNCTION  |  IF >            |  IN  |  LET   |  THEN  |  VAR       |  WHILE >            |  ASSIGN  as ":="  |  COLON   as ":"  |  COMMA   as ","   |  CPAREN  as ")" >            |  DIV     as "/"   |  EQU     as "="  |  LST     as "<="  |  MINUS   as "-" >            |  NEG     as "~"   |  OPAREN  as "("  |  PLUS    as "+"   |  SEMI    as ";" >            |  TIMES   as "*" >            |  ID   {String}    |  INT  {String};

A terminal symbol may carry a semantic value or attribute. The Haskell type of the semantic value is given in curly braces. As a rule, Haskell source code is always enclosed in curly braces within the grammar part. The as-clauses define shortcuts for terminals, which may then be used in the productions.

The declaration below introduces a nonterminal symbol called exp followed by sixteen productions for that symbol. The asterix marks exp as a start symbol; exp has a single attribute of type Expr.

> *exp  {Expr}; >  exp  {Var v}           :  ID {v}; >       {Block es}        |  "(", sepBy exp ";" {es}, ")"; >       {Int i   }        |  INT {i}; >       {Un Neg e}        |  "-", exp {e}, prec "~"; >       {Call f es}       |  ID {f}, "(", sepBy exp "," {es}, ")"; >       {Bin e1 Eq  e2}   |  exp {e1}, "=",  exp {e2}; >       {Bin e1 Leq e2}   |  exp {e1}, "<=", exp {e2}; >       {Bin e1 Add e2}   |  exp {e1}, "+",  exp {e2}; >       {Bin e1 Sub e2}   |  exp {e1}, "-",  exp {e2}; >       {Bin e1 Mul e2}   |  exp {e1}, "*",  exp {e2}; >       {Bin e1 Div e2}   |  exp {e1}, "/",  exp {e2}; >       {Assign v e}      |  ID {v}, ":=",  exp {e}; >       {IfThen e e1}     |  IF, exp {e}, THEN, exp {e1}; >       {IfElse e e1 e2}  |  IF, exp {e}, THEN, exp {e1}, ELSE, exp {e2}; >       {While e e1}      |  WHILE, exp {e}, DO, exp {e1}; >       {Let ds es}       |  LET, many dec {ds}, IN, sepBy exp ";" {es}, END;

Left-hand and right-hand side of a production are separated by a colon; symbols on the right are separated by commas and terminated by a semicolon. Alternative right-hand sides are separated by a vertical bar.

The pieces in curly braces constitute Haskell source code. Each rule can be seen as a function from the right-hand to the left-hand side. On the right-hand side, Haskell variables are used to name the values of attributes. The values of the attributes on the left-hand side are given by Haskell expressions, in which the variables of the right-hand side occur free.

The last production makes use of two (predefined) rule schemes: many x implements the repetition of the symbol x, and sepBy x sep denotes a repetition of x symbols separated by sep symbols.

The above productions are ambiguous as, for instance, 1 + 2 * 3 has two derivations. The ambiguity can be resolved by assigning precedences to terminal symbols.

>  left      7  "~";   left      6  "*";   left      6  "/";  left      5  "+";  left      5  "-"; >  right     0  THEN;  right     0  ELSE;   >  nonassoc  4  "<=";  nonassoc  4  "=";   nonassoc  0  DO;   nonassoc  0  ":=";

The following declarations define the nonterminal dec and three further nonterminals.

>  dec  {Decl}; >  dec  {d}  :  vardec {d}; >       {d}  |  fundec {d}; >  >  vardec  {Decl}; >  vardec  {Variable v e}  :  VAR, ID {v}, ":=", exp {e}; >  >  fundec  {Decl}; >  fundec  {Function f xs e}  :  FUNCTION, ID {f}, "(", sepBy (ID {}) "," {xs}, ")", "=", exp {e}; >  >  formal  {(Ident, TyIdent)}; >  formal  {(v, t)}  :  ID {v}, ":", ID {t}; >  >  }%

The grammar part ends here. The source file typically includes a couple of Haskell declarations. The user-defined function frown is the error routine invoked by the parser in case of a syntax error; its definition is mandatory.

>  frown _  =  error "syntax error" >  >  tc f  =  do {  putStrLn "*** reading ...";  s <- readFile f;     print s; >                 putStrLn "*** lexing  ...";  let {ts = lexer s};  print ts; >                 putStrLn "*** parsing ...";  e <- exp ts;         print e }

Figure 2.1: A sample Frown grammar file.

---

Now, type

The call frown -v Tiger.lg generates a Haskell parser which can then be loaded into hugs (or ghci). The parser has type exp :: (Monad m) => [Terminal] -> m Expr, that is, the parser is a computation that takes a list of terminals as input and returns an expression.

More examples can be found in the directory Manual/Examples:

Paren1.lg

well-balanced parentheses: a pure grammar (see Sec. 3.2.1);

Paren2.lg

an extension of Paren1.lg illustrating the definition of attributes (see Sec. 3.2.2);

Calc.lg

a simple evaluator for arithmetic expressions: a parser that interfaces with a separate lexer (see Sec. 3.2.3);

MCalc.lg

a variant of the desktop calculator (Calc.lg) that prints all intermediate results: illustrates monadic actions (see Sec. 3.2.4);

Let1.lg

an ambiguous expression grammar: illustrates backtracking parsers (see Sec. 3.2.5);

Let2.lg

an expression grammar: illustrates the use of precedences and associativity (see Sec. 3.2.6);

Let3.lg

a variant of the expression grammar: shows how to simulate inherited attributes using a reader monad (see Sec. 3.2.8);

Let4.lg

an expression grammar: illustrates a parser that interfaces with a monadic lexer (see Sec. 3.3.1);

Let5.lg

a variant of Let4.lg with better error reporting (see Sec. 3.3.2);

Let6.lg

a variant of Let5.lg with even better error reporting: prints a list of expected tokens upon error (see Sec. 3.3.3);

Let7.lg

yet another variant of the expression grammar: illustrates a simple form of error correction (see Sec. 3.3.4);

Let8.lg

variant of Let7.lg that notifies the user of corrections (see Sec. 3.3.4);

VarCalc.lg

a variant of the desktop calculator (Calc.lg) that works without a separate lexer: illustrates guards (see Sec. 3.4.2);

Paren3.lg

illustrates the tracing facilities (see Sec. 3.4.4);

VarParen.lg

illustrates irrefutable patterns on the right-hand side of productions (see Sec. 4.1);

RepMin.lg

a solution to the rep-min problem: illustrates how to simulate inherited attributes using functional attributes (see Sec. 4.2).

Chapter 3  Tour de Frown

This chapter introduces Frown by means of example.

3.1  Preliminaries: monads

Some elementary knowledge of monads is helpful in order to use Frown effectively. For the most basic applications, however, one can possibly do without. This section summarizes the relevant facts.

In Haskell, the concept of a monad is captured by the following class definition.

The essential idea of monads is to distinguish between computations and values. This distinction is reflected on the type level: an element of m a represents a computation that yields a value of type a. The trivial or pure computation that immediately returns the value a is denoted return a. The operator (>>=), commonly called ‘bind’, combines two computations: m >>= k applies k to the result of the computation m. The derived operation (>>) provides a handy shortcut if one is not interested in the result of the first computation. Finally, the operation fail is useful for signaling error conditions (a common thing in parsing).

Framing the concept of a monad as a type class is sensible for at least two interrelated reasons. First, we can use the same names (return, ‘>>=’, and fail) for wildly different computational structures.¹ Second, by overloading a function with the monad class we effectively parameterize the function by computational structures, that is, we can call the same function with different instances of monads obtaining different computational effects.

The following instance declaration (Result.lhs) defines a simple monad that we will use intensively in the sequel (the monad can be seen as a simplified term implementation of the basic monad operations).

In monad speak, this is an exception monad: a computation in Result either succeeds gracefully yielding a value a (represented by the term Return a) or it fails with an error message s (represented by Fail s). That’s all we initially need for Frown: parsing a given input either succeeds producing a semantic value (sometimes called an attribution) or it fails (hopefully, with a clear indication of the syntax error).

3.2  Basic features

3.2.1  Pure grammars

Let’s start with a simple example. The following complete Frown source file (Paren1.lg²) defines the language of well-balanced parentheses. The specification of the grammar is enclosed in special curly braces ‘%{ ldots }%’. The remainder contains Haskell source code, that is, a module header and a function declaration.

The part enclosed in special curly braces comprises the typical ingredients of a context-free grammar: a declaration of the terminal symbols, a declaration of the nonterminal symbols, and finally the productions or grammar rules.

In general, the terminal symbols are given by Haskell patterns of the same type. Here, we have two character patterns of type Char.

Nonterminals are just identifiers starting with a lower-case letter. By convention, the first nonterminal is also the start symbol of the grammar (this default can be overwritten, see Sec. 3.2.7).

Productions have the general form n : v_1, ldots, v_k; where n is a nonterminal and v_1, …, v_k are symbols. Note that the symbols are separated by commas and terminated by a semicolon. The mandatory trailing semicolon helps to identify so-called є-productions, productions with an empty right-hand side, such as paren   :   ;.

As a shorthand, we allow to list several alternative right-hand sides separated by a vertical bar. Thus, the above productions could have been written more succinctly as

The two styles can be arbitrarily mixed. In fact, it is not even required that productions with the same left-hand side are grouped together (though it is good style to do so).

Now, assuming that the above grammar resides in a file called Paren.g we can generate a Haskell parser by issuing the command

frown Paren.g

This produces a Haskell source file named Paren.hs that contains among other things the function

which recognizes the language generated by the start symbol of the same name. Specifically, if inp is a list of characters, then paren inp is a computation that either succeeds indicating that inp is a well-formed parentheses or fails indicating that inp isn’t well-formed. Here is a short interactive session using the Haskell interpreter Hugs (type hugs Paren.hs at the command line).

Note that we have to specify the result type of the expressions in order to avoid an unresolved overloading error. Or to put it differently, we have to specify the monad, in which the parsing process takes place. Of course, we are free to assign paren a more constrained type by placing an appropriate type signature in the Haskell section of the grammar file:

By the way, since the nonterminal paren carries no semantic value, the type of the computation is simply Result () where the empty tuple type ‘()’ serves as a dummy type. In the next section we will show how to add attributes or semantic values to nonterminals.

Every once in a while parsing fails. In this case, Frown calls a user-supplied function named, well, frown (note that you must supply this function). In our example, frown has type

The error function frown is passed the remaining input as an argument, that you can give an indication of the location of the syntax error (more on error reporting in Sec. 3.3). Note that frown must be polymorphic in the result type.

3.2.2  Attributes

Now, let’s augment the grammar of Sec. 3.2.1 by semantic values (Paren2.lg). Often, the parser converts a given input into some kind of tree representation (the so-called abstract syntax tree). To represent nested parentheses we simply use binary trees (an alternative employing n-ary trees is given in Sec. 4.1).

Attributes are always given in curly braces. When we declare a nonterminal, we have to specify the types of its attributes as in paren {Tree}. The rules of the grammar can be seen as functions from the right-hand side to the left-hand side. On the right-hand side, Haskell variables are used to name the values of attributes. The values of the attributes on the left-hand side are then given by Haskell expressions, in which the variables of the right-hand side may occur free. The Haskell expressions can be arbitrary, except that they must not be layout-sensitive.

In general, a nonterminal may have an arbitrary number of attributes (see Sec. 4.4 for an example). Note that Frown only supports so-called synthesized attributes (inherited attributes can be simulated, however, with the help of a reader monad, see Sec. 3.2.8, or with functional attributes, see Sec. 4.2).

The parser generated by Frown now has type

The following interactive session illustrates its use.

3.2.3  Interfacing with a lexer

The parsers of the two previous sections take a list of characters as input. In practice, a parser usually does not work on character streams directly. Rather, it is prefaced by a lexer that first converts the characters into a list of so-called tokens. The separation of the lexical analysis from the syntax analysis usually leads to a clearer design and as a benevolent side-effect it also improves efficiency (Sec. 3.4.2 shows how to combine lexing and parsing in Frown, though).

A simple token type is shown in Fig 3.3 (Terminal1.lhs). (Note that the type comprises more constructors than initially needed.)

---

>  module Terminal where >  import Maybe >  >  data Op                       =   Plus | Minus | Times | Divide >                                    deriving (Show) >  >  name                          ::  Op -> String >  name Plus                     =   "+" >  name Minus                    =   "-" >  name Times                    =   "*" >  name Divide                   =   "/" >  >  app                           ::  Op -> (Int -> Int -> Int) >  app Plus                      =   (+) >  app Minus                     =   (-) >  app Times                     =   (*) >  app Divide                    =   div >  >  data Terminal                 =   Numeral Int >                                |   Ident String >                                |   Addop Op >                                |   Mulop Op >                                |   KWLet >                                |   KWIn >                                |   Equal >                                |   LParen >                                |   RParen >                                |   EOF >                                    deriving (Show) >  >  ident, numeral                ::  String -> Terminal >  ident   s                     =   fromMaybe (Ident s) (lookup s keywords) >  numeral s                     =   Numeral (read s) >  >  keywords                      ::  [(String, Terminal)] >  keywords                      =   [ ("let", KWLet), ("in", KWIn) ]

Figure 3.3: The type of terminals (Terminal1.lhs).

---

Fig. 3.4 (Lexer.lhs) displays a simple lexer for arithmetic expressions, which are built from numerals using the arithmetic operators ‘+’, ‘-’, ‘*’, and ‘/’.

---

>  module Lexer  (module Terminal, module Lexer) where >  import Char >  import Terminal >  >  lexer                         ::  String -> [Terminal] >  lexer []                      =   [] >  lexer (’+’  : cs)             =   Addop Plus    : lexer cs >  lexer (’-’  : cs)             =   Addop Minus   : lexer cs >  lexer (’*’  : cs)             =   Mulop Times   : lexer cs >  lexer (’/’  : cs)             =   Mulop Divide  : lexer cs >  lexer (’=’  : cs)             =   Equal         : lexer cs >  lexer (’(’  : cs)             =   LParen        : lexer cs >  lexer (’)’  : cs)             =   RParen        : lexer cs >  lexer (c : cs) >      | isAlpha c               =   let (s, cs’) = span isAlphaNum  cs  in  ident    (c : s) : lexer cs’ >      | isDigit c               =   let (s, cs’) = span isDigit     cs  in  numeral  (c : s) : lexer cs’ >      | otherwise               =   lexer cs

Figure 3.4: A simple lexer for arithmetic expressions (Lexer.lhs).

---

The following grammar, which builds upon the lexer, implements a simple evaluator for arithmetic expressions (Calc.lg).

The terminal declaration now lists patterns of type Terminal. Note that terminals may also carry semantic values. The single argument of Numeral, for instance, records the numerical value of the numeral.

When declaring a terminal we can optionally define a shortcut using an as-clause as, for example, in LParen as "(". The shortcut can be used in the productions possibly improving their readability.

Here is an example session demonstrating the evaluator.

3.2.4  Monadic actions

The expression that determines the value of an attribute is usually a pure one. It is, however, also possible to provide a monadic action that computes the value of the attribute. The computation lives in the underlying parsing monad. Monadic actions are enclosed in ‘{% ldots }’ braces and have type m t where m is the type of the underlying monad and t is the type of attributes.

As an example, the following variant of the desktop calculator (MCalc.lg) prints all intermediate results (note that we only list the changes to the preceeding example).

The following session illustrates its working.

In general, monadic actions are useful for performing ‘side-effects’ (for example, in order to parse %include directives) and for interaction with a monadic lexer (see Sec. 3.3.1).

3.2.5  Backtracking parsers

In the previous examples we have encoded the precedences of the operators (‘*’ binds more tightly than ‘+’) into the productions of the grammar. However, this technique soon becomes unwieldy for a larger expression language. So let’s start afresh. The grammar file shown in Fig. 3.5 (Let1.lg) uses only a single nonterminal for expressions (we have also extended expressions by local definitions).

---

>  module Let where >  import Lexer >  import Monad >  >  data Expr                     =  Const Int | Var String | Bin Expr Op Expr | Let Decl Expr >                                   deriving (Show) >  >  data Decl                     =  String :=: Expr >                                   deriving (Show) >  >  %{ >  >  Terminal                      =  Numeral {Int} >                                |  Ident {String} >                                |  Addop {Op} >                                |  Mulop {Op} >                                |  KWLet   as "let" >                                |  KWIn    as "in" >                                |  Equal   as "=" >                                |  LParen  as "(" >                                |  RParen  as ")"; >  >  expr  {Expr}; >  expr  {Const n}               :  Numeral {n}; >        {Var s}                 |  Ident {s}; >        {Bin e1 op e2}          |  expr {e1}, Addop {op}, expr {e2}; >        {Bin e1 op e2}          |  expr {e1}, Mulop {op}, expr {e2}; >        {Let d e}               |  "let", decl {d}, "in", expr {e}; >        {e}                     |  "(", expr {e}, ")"; >  >  decl  {Decl}; >  decl  {s :=: e}               :  Ident {s}, "=", expr {e}; >  >  }% >  >  frown _                       =  fail "syntax error"

Figure 3.5: An ambiguous grammar (Let1.lg).

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Also note that the grammar has no Nonterminal declaration. Rather, the terminal symbols are declared by supplying type signatures before the respective rules. Generally, type signatures are preferable to a Nonterminal declaration if the grammar is long.

Of course, the rewritten grammar is no longer LALR(k) simply because it is ambiguous. For instance, ‘1+2*3’ can be parsed as Bin (Const 1) Plus (Bin (Const 2) Times (Const 3)) or as Bin (Bin (Const 1) Plus (Const 2)) Times (Const 3). Frown is also unhappy with the grammar: it reports six shift/reduce conflicts:

This means that Frown wasn’t able to produce a deterministic parser. Or rather, it produced a deterministic parser by making some arbitrary choices to avoid non-determinism (shifts are preferred to reductions, see Sec. 3.2.6). However, we can also instruct Frown to produce a non-deterministic parser, that is, one that generates all possible parses of a given input. We do so by supplying the option --backtrack:

frown --backtrack Let.g

The generated parser expr now has type

Note that the underlying monad must be an instance of MonadPlus (defined in the standard library Monad). The list monad and the Maybe monad are both instances of MonadPlus. The following session shows them in action.

The list monad supports ‘deep backtracking’: all possible parses are returned (beware: the number grows exponentionally). The Maybe monad implements ‘shallow backtracking’: it commits to the first solution (yielding the same results as the parser generated without the option --backtrack).

3.2.6  Precedences and associativity

Instead of resorting to a backtracking parser we may also help Frown to generate the ‘right’ deterministic parser by assigning precedences to terminal symbols. The understand the working of precedences it is necessary to provide some background of the underlying parsing technique.

LR parsers work by repeatedly performing two operations: shifts and reductions. A shift moves a terminal from the input onto the stack, the auxiliary data structure maintained by the parser. A reduction replaces a top segment of the stack matching the right-hand side of a production by its left-hand side. Parsing succeeds if the input is empty and the stack consists of a start symbol only. As an example, consider parsing ‘N*N+N’.

At this point, there are two possibilities: we can either perform a reduction (using the production e : e, *, e;) or shift the next input symbol. Both choices are viable.

Alas, the two choices also result in different parse trees. By default, Frown prefers shifts to reductions. As a consequence, N*N+N is parsed as N*(N+N), that is, ‘+’ binds more tightly than ‘*’.

Now, we can direct the resolution of conflicts by assigning precedences and associativity to terminal symbols. The following declarations will do in our example (Let2.g).

Thus, ‘*’ takes precedence over ‘+’, which in turn binds more tightly than ‘in’. For instance, let a = 4 in a + 2 is parsed as let a = 4 in (a + 2). A conflict between two symbols of equal precedence is resolved using associativity: the succession 1+2+3 of left-associative operators is grouped as (1+2)+3; likewise for right-associative operators; sequences of non-associative operators are not well-formed.

Given the fixity declarations above Frown now produces the ‘right’ deterministic parser, which can be seen in action below.

In general, a conflict between the actions ‘reduce by rule r’ and ‘shift terminal t’ is resolved as follows (the precedence of a rule is given by the precedence of the rightmost terminal on the right-hand side):

Just in case you may wonder: there are no shift/shift conflicts by construction; reduce/reduce conflicts cannot be cured using precedences and associativity.

3.2.7  Multiple start symbols

A grammar may have several start symbols. In this case, Frown generates multiple parsers, one for each start symbol (actually, these are merely different entry points into the LR(0) automaton⁴). We mark a symbol as a start symbol simply by putting an asterix before its declaration (either in a Nonterminal declaration or in a separate type signature). Consider our previous example: most likely we want parsers both for expressions and declarations. Thus, we write

and get parsers of type.

3.2.8  Monadic attributes

This section does not introduce any new features of Frown and can be safely skipped on first reading. Its purpose is to show how to simulate inherited attributes using a reader monad (see also Sec. 4.2). Generally, inherited attributes are used to pass context information down the parse tree. As an example, consider implementing an evaluator for arithmetic expressions that include variables and let-bindings (Let3.lg). To determine the value of a variable we need to pass down an environment that records the values of bound variables. The reader monad displayed in Fig. 3.6 (Reader.lhs) serves exactly this purpose.

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>  module Reader where >  >  newtype Reader env a          =   Reader { apply :: env -> a } >  >  instance Monad (Reader env) where >      return a                  =   Reader (\ env -> a) >      m >>= k                   =   Reader (\ env -> apply (k (apply m env)) env) >      fail s                    =   Reader (error s) >  >  getenv                        ::  Reader env env >  getenv                        =   Reader (\ env -> env) >  >  withenv                       ::  env -> Reader env a -> Reader env’ a >  withenv env m                 =   Reader (\ env’ -> apply m env)

Figure 3.6: The reader monad (Reader.lhs).

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We need some additional helper functions for accessing and extending environments

The following grammar implements the desired evaluator.

Note that there are two monads around: the parsing monad (in fact, expr is parametric in this monad) and the reader monad, which is embedded in the attributes. The parser returns a value of type Reader Int, to which we pass an empty initial environment.

Let’s see the evaluator in action.

3.3  Error reporting and correction

3.3.1  Monadic lexers

The chances that parsing succeeds are probably smaller than the chances that it fails. Good error messages are indispensable to turn the latter into the former case. Up to now we only produced the rather uninformative message "syntax error". Fortunately, we are in a good position to do better. LR parsing has the nice property that it detects a syntax error at the earliest possible moment: parsing fails as soon as the input cannot be extended to a legal sentence of the grammar. For instance, the syntax error in let  a = 4 * (7 + 1 − 1  in a * a is detected after reading the keyword ‘in’.

Now, all we have to do is to keep track of context information: the current line and column number and possibly the filename. This section prepares the ground for maintaining state information; the parser that actually keeps track of line numbers etc is only shown in the next section.

Unsurprisingly, to maintain state information we employ monads again. This time, we require a state monad. The natural place for maintaining information about line numbers etc is, of course, the lexer. Consequently, we turn the stream-based lexer of type String -> [Terminal] into a monadic one of type

where M is the state monad. The idea is that each time get is called it returns the next token and updates its internal state.

The first version of the monadic lexer shown in Fig. 3.7 (MLexer1.lhs) has no internal state apart from the input stream, that is, it provides no additional functionality compared to the stream-based lexer.

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>  module MLexer ( module Terminal, module MLexer ) where >  import Terminal >  import Char >  >  type CPS a answer             =   (a -> answer) -> answer >  >  newtype Lex m a               =   Lex { unLex :: forall ans . CPS a (String -> m ans) } >  >  instance (Monad m) => Monad (Lex m) where >      return a                  =   Lex (\ cont -> cont a) >      m >>= k                   =   Lex (\ cont -> unLex m (\ a -> unLex (k a) cont)) >      fail s                    =   lift (fail s) >  >  lift                          ::  (Monad m) => m a -> Lex m a >  lift m                        =   Lex (\ cont inp -> m >>= \ a -> cont a inp) >  >  run                           ::  (Monad m) => Lex m a -> (String -> m a) >  run parser inp                =   unLex parser (\ a rest -> return a) inp >  >  get                           ::  (Monad m) => Lex m Terminal >  get                           = >    Lex (\ cont inp -> >      let lexer []              =   cont (EOF)          [] >          lexer (’+’  : cs)     =   cont (Addop Plus)   cs >          lexer (’-’  : cs)     =   cont (Addop Minus)  cs >          lexer (’*’  : cs)     =   cont (Mulop Times)  cs >          lexer (’/’  : cs)     =   cont (Mulop Divide) cs >          lexer (’=’  : cs)     =   cont (Equal)        cs >          lexer (’(’  : cs)     =   cont (LParen)       cs >          lexer (’)’  : cs)     =   cont (RParen)       cs >          lexer (c : cs) >            | isSpace c         =   lexer cs >            | isAlpha c         =   let (s, cs’) = span isAlphaNum  cs in cont (ident    (c : s)) cs’ >            | isDigit c         =   let (s, cs’) = span isDigit     cs in cont (numeral  (c : s)) cs’ >            | otherwise         =   lexer cs >      in  lexer inp) >  >  frown                         ::  (Monad m) => Terminal -> Lex m a >  frown t                       =   Lex (\ cont inp -> >                                      fail ("\n*** syntax error:\n" ++ context 4 inp)) >  >  context                       ::  Int -> String -> String >  context n inp                 =   unlines (take n (lines inp ++ ["<end of input>"]))

Figure 3.7: A monadic lexer for the let language (MLexer1.lhs).

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Note that we use a continuation-based state monad, Lex m, which requires local universal quantification (a non-Haskell 98 feature). Actually, Lex is even a monad transformer so that we can freely choose a base monad (such as Result or IO). Of course, an ‘ordinary’ state monad would do, as well. The monadic lexer get incorporates more or less the stream-based lexer. We only changed the recursive calls to lexer (ie t : lexer cs) into invocations of the continuation (ie cont t cs). The error routine frown now has type

that is, frown is no longer passed the remaining input but only the look-ahead token.

The changes to the grammar are minor: we have to declare an ‘end of file’ token marked by a star (Let4.lg)

and we have to provide a type signature for the generated parser (in the Haskell section).

The type signature is necessary to avoid an ‘unresolved top-level overloading’ error (the monomorphism restriction strikes again).

When we generate the Haskell parser we must supply the option --lexer to inform Frown that we use a monadic lexer.

frown --lexer Let.g

For completeness, here is an interactive session (note that in the case of error the look-ahead token is not displayed).

3.3.2  Error reporting

The monadic lexer shown in Fig. 3.8 (MLexer2.lhs) builds upon the one given in the previous section. The state monad Lex m has been extended to keep track of the current line number and the current line itself. The current line is displayed in case of a lexical or syntax error. As an aside, note that the column number can be recreated from the rest of the input and the length of the current line.

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>  module MLexer ( module Terminal, module MLexer ) where >  import Terminal >  import Char >  >  type CPS a answer             =   (a -> answer) -> answer >  >  newtype Lex m a               =   Lex { unLex :: forall ans . CPS a (String -> Int -> String -> m ans) } >  >  instance (Monad m) => Monad (Lex m) where >      return a                  =   Lex (\ cont -> cont a) >      m >>= k                   =   Lex (\ cont -> unLex m (\ a -> unLex (k a) cont)) >      fail s                    =   lift (fail s) >  >  lift                          ::  (Monad m) => m a -> Lex m a >  lift m                        =   Lex (\ cont inp line cur -> m >>= \ a -> cont a inp line cur) >  >  run                           ::  (Monad m) => Lex m a -> (String -> m a) >  run parser inp                =   unLex parser (\ a rest line cur -> return a) inp 1 (current inp) >  >  current                       ::  String -> String >  current s                     =   takeWhile (/= ’\n’) s >  >  get                           ::  (Monad m) => Lex m Terminal >  get                           = >   Lex (\ cont inp line cur -> >    let  lexer []          n x  =   cont (EOF)          [] n       x >         lexer (’\n’ : cs) n x  =   lexer               cs (n + 1) (current cs) >         lexer (’+’  : cs) n x  =   cont (Addop Plus)   cs n       x >         lexer (’-’  : cs) n x  =   cont (Addop Minus)  cs n       x >         lexer (’*’  : cs) n x  =   cont (Mulop Times)  cs n       x >         lexer (’/’  : cs) n x  =   cont (Mulop Divide) cs n       x >         lexer (’=’  : cs) n x  =   cont (Equal)        cs n       x >         lexer (’(’  : cs) n x  =   cont (LParen)       cs n       x >         lexer (’)’  : cs) n x  =   cont (RParen)       cs n       x >         lexer (c : cs)    n x >           | isSpace c          =   lexer cs n x >           | isAlpha c          =   let (s, cs’) = span isAlphaNum  cs in cont (ident    (c : s)) cs’ n x >           | isDigit c          =   let (s, cs’) = span isDigit     cs in cont (numeral  (c : s)) cs’ n x >           | otherwise          =   fail ("\n*** lexical error at " >                                          ++ position cs n x ++ ":\n" >                                          ++ context 4 cs x) >    in  lexer inp line cur) >  >  frown                         ::  (Monad m) => Terminal -> Lex m a >  frown t                       =   Lex (\ cont inp line cur -> >                                      fail ("\n*** syntax error at " >                                            ++ position inp line cur ++ ":\n" >                                            ++ context 4 inp cur)) >  >  position                      ::  String -> Int -> String -> String >  position inp line cur         =   "(line "  ++ show line ++ ", column " ++ show col ++ ")" >      where col                 =   length cur - length (current inp) >  >  context                       ::  Int -> String -> String -> String >  context n inp cur             =   unlines ([cur, replicate col’ ’ ’ ++ "^"] >                                             ++ take n (lines (drop 1 (dropWhile (/= ’\n’) inp)) >                                                        ++ ["<end of input>"])) >      where col’                =   length cur - length (current inp) - 1

Figure 3.8: A monadic lexer for the let language featuring good error reports (MLexer2.lhs).

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The following session shows the new lexer in action.

In the case of a lexical error the cursor ‘^’ points at the offending character. In the case of a syntax error the cursor points at the last character of the offending token (recall that the part of the input up to and including this token is the shortest prefix of the input that cannot be extended to a legal sentence of the grammar).

3.3.3  Expected tokens

We can do even better! We can instruct Frown to pass a list of expected tokens to the error routine frown (by supplying the option --expected).

frown --lexer --expected Let.g

Frown uses the shortcuts given in the terminal declaration for generating lists of expected tokens. This means, in particular, that a token is not included in such a list if it does not have a shortcut. In our running example, we want every token to be listed. Therefore, we add shortcuts for every terminal symbol (Let6.lg).

The error routine frown now takes an additional argument of type [String] (MLexer3.lhs).

The interactive session listed in Fig. 3.9 is a bit longer than usual to illustrate the quality of the error messages.

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>  Let>> run expr "let\n    a = 4 * [7 + 1 - 1)\n in a * a" :: IO Expr >   >  Program error: user error ( >  *** lexical error at (line 2, column 13): >      a = 4 * [7 + 1 - 1) >              ^ >   in a * a >  <end of input> >  ) >  Let>> run expr "let\n    a = 4 * (7 + 1 - 1\n in a * a" :: IO Expr >   >  Program error: user error ( >  *** syntax error at (line 3, column 3): >   in a * a >    ^ >  <end of input> >  * expected: + or -, * or /, )) >  Let>> run expr "let\n    a = 4 * (7 + 1 - 1)\n a * a" :: IO Expr >   >  Program error: user error ( >  *** syntax error at (line 3, column 2): >   a * a >   ^ >  <end of input> >  * expected: + or -, * or /, in) >  Let>> run expr "\n    a = 4 * (7 + 1 - 1)\n in a * a" :: IO Expr >   >  Program error: user error ( >  *** syntax error at (line 2, column 7): >      a = 4 * (7 + 1 - 1) >        ^ >   in a * a >  <end of input> >  * expected: + or -, * or /, <end of input>) >  Let>> run expr "let\n    a = 4 * (7 + - 1)\n in a * a" :: IO Expr >   >  Program error: user error ( >  *** syntax error at (line 2, column 18): >      a = 4 * (7 + - 1) >                   ^ >   in a * a >  <end of input> >  * expected: <numeral>, <identifier>, let, () >  Let>> run expr "let\n    a = 4  (7 + 1 - 1)\n in a * a" :: IO Expr >   >  Program error: user error ( >  *** syntax error at (line 2, column 12): >      a = 4  (7 + 1 - 1) >             ^ >   in a * a >  <end of input> >  * expected: + or -, * or /, in)

Figure 3.9: A session full of syntax errors.

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3.3.4  Error correction

So far we have content ourselves with reporting syntax errors. To a limited extent it is also possible to correct errors. Consider the last rule of the following grammar (Let7.lg).

The symbol insert ")" instructs Frown to automatically insert a ")" token if parsing would otherwise fail. The special symbol insert ")" can be seen as being defined by the є-production insert ")"   :  ;. The difference to an ‘ordinary’ user-defined є-production is that the rule is only applied if every other action would fail.

The following session shows the error correction in action.

In the last query the missing parenthesis ‘)’ is inserted just before the keyword ‘in’. This may or may not what the user intended!

It is generally a good idea to notify the user if a token is inserted. This is relatively easy to accomplish using monadic actions (Let8.lg). The parsing monad is now Lex IO; the monad transformer Lex proves its worth.

Let’s repeat the last query of the previous session.

The reader is invited to extend the code so that the current source location is additionally printed (informing the user where the token has been inserted).

3.4  Advanced features

3.4.1  Rule schemes

When we define grammars we often find ourselves repeatedly writing similar rules. A common pattern is the repetition of symbols. As an example, the following rules define a repetition of t symbols.

As an aside, note that the second rule is intentionally left-recursive. LR parsers prefer left to right recursion: the rules above use constant stack space whereas the right-recursive variant requires space linear in the length of the input.

Now, Frown allows to capture recurring patterns using so-called rule schemes. Here is the scheme for a repetition of symbols (of arity 0).

The first line contains many’s type signature: it simply says that neither many nor many’s argument x possess attributes. Given this scheme we can simply write many t for a repetition of t symbols.

The rule for repetition becomes more interesting if the argument possesses an attribute (is of arity 1). In this case, many returns a list of semantic values.

(The use of list concatenation ‘++’ in the second rule incurs a runtime penalty which we will cure later.) The first line contains again the type signature, which we may read as a conditional clause: if x has one attribute of type a, then many x has one attribute of type [a]. This schemes comes in handy if we extend our little expression language by applications and abstractions (we assume that the abstract syntax has been extended suitably; aexpr denotes atomic expressions).

Note that if we pass terminal symbols as arguments to rule schemes they must be written with (empty) curly braces—Frown can only identify terminal symbols, ie patterns, if they have exactly the same syntactic form as in the terminal declaration. Think of ‘{}’ as a placeholder.

In the above definition of many we have used list concatenation to append an element to a list. The following improved definition does away with this linear-time operation employing Hughes’ efficient sequence type [3].

These schemata are predefined in Frown. There is a caveat, however: the singleton production many x   :   many’ x may introduce a shift/reduce conflict, see Sec. 4.3.

Actually, both the many scheme with no attributes and the scheme above with one attribute are predefined. In general, it is possible to use the same name for schemes and nonterminals of different arity. The only restriction is that the arity of the scheme must determine the arity of its arguments.

Another useful variation of many is sepBy x sep which denotes a list of x symbols separated by sep symbols (sepBy and sepBy1 are predefined, as well).

This scheme is useful for adding tuples to our expression language.

For a complete list of predefined schemes see Sec. 5.3.

3.4.2  A second look at terminal symbols

The terminal symbols of a grammar are given by Haskell patterns. Up to now we have seen only simple patterns. Patterns, however, may also be nested or even overlapping. In the latter case, one should be careful to list specific patterns before general ones in a Terminal declaration (Frown preserves the relative ordering of patterns when generating case expressions). Here is a simple example.

Note that keywords are declared just by listing them before the general pattern for identifiers.

Alternatively, terminal symbols can be specifed using so-called guards, Boolean functions of type Terminal -> Bool. Guards are most useful for defining character classes as in the following example.

A guard is introduced by the keyword guard, followed by its Haskell definition, followed by the mandatory shortcut. The shortcut can then be used as a terminal symbol of arity 1: its attribute of type Terminal is the very input symbol that matched the guard.

Using guards one can quite easily define character-based grammars that include lexical syntax (that is, whose parsers combine lexing and parsing). Fig. 3.10 lists a variant of the desktop calculator that works without a separate lexer.

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>  module Calc where >  import Result >  import Char >  >  type Terminal                 =  Char >  >  %{ >  >  Terminal                      =  guard {isDigit}  as "digit" >                                |  ’+’ >                                |  ’*’ >                                |  ’(’ >                                |  ’)’; >  >  Nonterminal                   =  expr     {Integer} >                                |  term     {Integer} >                                |  factor   {Integer} >                                |  numeral  {Integer}; >  >  expr     {v1 + v2}            :  expr {v1},  ’+’,  term {v2}; >           {e}                  |  term {e}; >  term     {v1 * v2}            :  term {v1},  ’*’,  factor {v2}; >           {e}                  |  factor {e}; >  factor   {n}                  :  numeral {n}; >           {e}                  |  ’(’, expr {e}, ’)’; >  numeral  {encode c}           :  "digit" {c}; >           {n * 10 + encode c}  |  numeral {n}, "digit" {c};  >  >  }% >  >  encode c                      =  toInteger (fromEnum  c - fromEnum ’0’) >  >  frown _                       =  fail "syntax error"

Figure 3.10: A variant of the desktop calculator that includes lexical syntax (VarCalc.lhs).

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Note that the type Terminal must be defined in the Haskell section. The reader may wish to extend the grammar so that two tokens can be separated by white space.

3.4.3  Look-ahead

⟨To do: type grammar.⟩

3.4.4  Debugging and tracing

⟨To do: --prefix und --suffix.⟩

⟨To do: --debug und --pagewidth.⟩